PID Tuning Calculator: Calculate Kp, Ki and Kd

Updated: August 28, 2026

PID tuning can be one of the most difficult parts of building a feedback control system. Even when your PID code is correct, the controller will not perform well until the proportional, integral, and derivative gains are properly adjusted.

This PID tuning calculator uses the classic Ziegler-Nichols closed-loop method to calculate starting values for Kp, Ki, and Kd. Enter the ultimate gain and oscillation period of your system, choose the controller type, and the calculator will generate the corresponding tuning constants.

How to Use the PID Tuning Calculator

The calculator requires two values measured from your system:

• Ku (Ultimate Gain) — the proportional gain at which the system produces sustained oscillation.
• Tu (Ultimate Period) — the time required for one complete oscillation at the ultimate gain.

You can enter the oscillation period in either seconds or milliseconds.

Then select whether you want a P, PI, or PID controller and click Calculate PID Values.

The calculator returns the recommended starting values for:

• Kp — proportional gain
• Ki — integral gain
• Kd — derivative gain

For Arduino users, the tool also generates the corresponding constants for the popular PID_v1 library.

Finding Ku and Tu

To use the Ziegler-Nichols closed-loop tuning method, first disable the integral and derivative terms:

Ki = 0;
Kd = 0;

Start with a low Kp value and gradually increase it while observing the system response.

At some gain, the output may begin oscillating around the setpoint without the oscillation continuously growing or disappearing. This gain is called the ultimate gain, Ku.

Measure the time between two consecutive peaks of the oscillation. This is the ultimate period, Tu.

For example, suppose sustained oscillation occurs when:

Ku = 4.0
Tu = 2.0 seconds

For a PID controller, the classic Ziegler-Nichols equations produce approximately:

Kp = 2.4
Ki = 2.4
Kd = 0.6

These values can then be used as the starting point for further tuning.

[IMAGE PLACEHOLDER: Graph showing sustained oscillation with Ku and Tu labeled]

Ziegler-Nichols PID Formulas

The calculator uses the classic closed-loop Ziegler-Nichols rules.

For a proportional controller:

$K_p = 0.5K_u$

For a PI controller:

$K_p = 0.45K_u$

$K_i = \frac{K_p}{T_u/1.2}$

For a PID controller:

$K_p = 0.6K_u$

$K_i = \frac{K_p}{T_u/2}$

$K_d = K_p\frac{T_u}{8}$

These equations provide a systematic way to obtain initial PID parameters instead of choosing arbitrary Kp, Ki, and Kd values.

Using the Results with Arduino

If you are using Brett Beauregard's Arduino PID library, the calculated values can be inserted directly into your sketch:

double Kp = 2.4;
double Ki = 2.4;
double Kd = 0.6;

PID myPID(&Input, &Output, &Setpoint,
          Kp, Ki, Kd, DIRECT);

The calculator automatically generates a similar code snippet using your calculated values.

For a complete explanation of Arduino PID control, including manual PID implementation, the PID_v1 library, integral windup, sample time, and practical tuning, see my Arduino PID Controller Tutorial.

The Calculated Values Are Starting Points

Ziegler-Nichols tuning does not guarantee the best possible response for every system.

The method tends to produce relatively aggressive tuning and may result in noticeable overshoot. Some systems also should not be deliberately driven into sustained oscillation because doing so could damage a motor, heater, mechanical assembly, or other hardware.

After calculating Kp, Ki, and Kd, observe how your actual system responds.

Typical adjustments include:

The goal of this PID calculator is therefore not to produce a universally perfect controller, but to give you a useful set of initial PID tuning values that you can refine on your actual hardware.

PID Tuning Calculator Summary

To use the calculator:

1. Set Ki and Kd to zero.
2. Increase Kp until sustained oscillation occurs.
3. Record this gain as Ku.
4. Measure the oscillation period Tu.
5. Enter Ku and Tu into the calculator.
6. Select P, PI, or PID control.
7. Use the calculated values as your starting point.
8. Fine-tune the controller while observing the real system response.

PID tuning is ultimately an experimental process. The calculator gives you a more systematic place to start, while testing on the actual system determines the final Kp, Ki, and Kd values.